Noun

(en noun)

*{{quote-magazine, date=2013-08-10, volume=408, issue=8848, magazine=(The Economist)

, title= A new prescription , passage=As the world's drug habit shows, governments are failing in their quest to monitor every London window-box and Andean hillside for banned plants. But even that Sisyphean task looks easy next to the fight against synthetic drugs. No sooner has a drug been blacklisted than chemists adjust their recipe and start churning out a subtly different one.}}

Any set of instructions for preparing a mixture of ingredients.

By extension, a plan or procedure to obtain a given end result.

*{{quote-magazine, date=2014-06-21, volume=411, issue=8892, magazine=(The Economist)

, title= Magician’s brain , passage=[Isaac Newton] was obsessed with alchemy. He spent hours copying alchemical recipes and trying to replicate them in his laboratory. He believed that the Bible contained numerological codes. The truth is that Newton was very much a product of his time.}}

Now especially, a set of instructions for making or preparing food dishes.

A set of conditions and parameters of an industrial process to obtain a given result.

Noun

(en noun)

A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.

Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny -

(mathematics, order theory) A non-empty]] lower set (of a partially ordered set) which is [[closure, closed under binary suprema (a.k.a. joins).^{[http://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem#Prime_ideal_theorems]}

If (1) the empty set were called a "small" set, and (2) any subset of a "small" set were also a "small" set, and (3) the union of any pair of "small" sets were also a "small" set, then the set of all "small" sets would form an ideal .

(for example, algebra) A subring closed under multiplication by its containing ring.

Let $\backslash mathbb\{Z\}$ be the ring of integers and let $2\backslash mathbb\{Z\}$ be its ideal of even integers. Then the quotient ring $\backslash mathbb\{Z\}\; /\; 2\backslash mathbb\{Z\}$ is a Boolean ring.

The product of two ideals $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$ is an ideal $\backslash mathfrak\{a\; b\}$ which is a subset of the intersection of $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$. This should help to understand why maximal ideals' are prime ' ideals . Likewise, the union of $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$ is a subset of $\backslash mathfrak\{a\; +\; b\}$.